When simplifying expression we collect like terms so that the expression is reduced to it's simplest form.
Example:
x + x + y = 2x + y
x2 + 2x + 3x + 12 = x2 + 5x + 12
We can't collect terms x2 and x together in the same way that we can't add together an area and a length.
When expanding out brackets we need to make sure that the term outside of the bracket is multiplied by everything inside.
a(b + c) = ab + ac This is called the distributive law.
When expanding out double brackets we need to apply the same principle.
(a + b)(c + d) = a(c + d) + b(c + d)
= ac + ad + bc + bd
If the values of a,b,c and d allow then terms can be collected togther to simplify the expression.
There are 2 methods here than I like to use:
FOIL Method - remembering to multiply the First terms, Outside terms, Inside terms and Last terms in the double brackets.
E.g. (a + b)(c + d)
F = ac
O = ad
I = bc
L = bd
Smiley Face Method - create 2 eyebrows, and nose and a mouth to join all the terms, then Multiply along the lines to expand out the brackets.
E.g. (a + b)(c + d)
left eyebrow = ac
right eyebrow = bd
nose = bc
mouth = ad
Solving Linear Equations
An equation is an expression which is equal to something. Example 1
x + 6 = 9
Think of an equation like balanced scales. When we start the equation the scales are balanced. If we are to change anything on one side of the equals sign, we must always do the same thing to the other side for the scales to remain balanced.
x + 6 = 9 (to get x on it's own we want to take 6 away on the LHS, but we must remember to do the same to the RHS)
x + 6 - 6 = 9 - 6
x = 3
When solving an equation, we always line up the equal signs. The practise of Maths is an art and we should respect that in our working out. I just can't get it to line up right on this wiki!
Example 2
4x = 20 (we divide both sides of the equation by 4) 4x = 20
4 4
x = 5
If I went into a shop and bought 6 twix's and a 10p mix for £1.90, how much did each twix cost?
Re write this as an equation to solve it. [twix = t]
6t + 0.10 = 1.90 (firstly we subtract 10p from both sides)
6t + 0.10 - 0.10 = 1.90 - 0.10
6t = 1.80 (then we divide both side by 6) 6t = 1.80
6 = 6
t = 0.30
We started with the month you were born, lets call this m. Then we multiplied by 5
Then we added 7 to our answer
Then we multiplied our answer by 4
Then we added 13 to our answer
Then we multiplied by 5
Then we added the day number of our birthday
m
5m
5m + 7
4(5m + 7) = 20m + 28
20m + 28 + 13 = 20m + 41
5(20m + 41) = 100m + 205
100m + 205 + d = 100m + d + 205
Giving the formula 100m + d + 205
Say we are looking at the 4th of July:
100(July) + 4 + 205 = 100(7) + 4 + 205 704 + 205
Can you see how the trick will always work?
If you give me the number 909
I subtract 205 to give me 704 where 04 is the day and 7 is the month!
Try is out on your friends – can you amaze them with you mathmagic abilities?!
When we have tricky equations that look like this:
x + 1 = 7
2
It's not as tricky as you first might think... but make sure you follow the steps and what you do to one side you must do to the other to make sure that the balance stays equal!
This equation is the same as 1/2(x + 1) = 7
We can either expand the brackets out - or to make it easier, as (x+1) is all multiplied by 1/2 we can multiply by 2 to get x + 1 on it's own on the LHS, like this.
1/2(x + 1) = 7
x + 1 = 2 x 7
x + 1 = 14 (now to get x on it's own to solve it we subtract 1 from both sides)
x = 14 - 1 x = 13
We've cracked it! - It's not as scary as it first looks
Give it a try with some of the questions from the lesson and see how well you can solve equations.
Solving Equations often relies on methods you have already learned about simplifying expressions, expanding brackets and collecting like terms.
Example:
2(8x - 7) = 34 Method 1
16x - 14 = 34 (expand out the brackets first)
16x = 48 (add 14 to both sides) x = 3 (divide both sides by 16)
Method 2
8x - 7 = 17 (divide both sides by 2- we can do this as 2 is multiplied by everything inside the bracket on the left hand side)
8x = 24 (add 7 to both sides) x = 3 (divide both sides by 8)
Try both method to see that they both work!
What about when we have an unknown on both sides?
Example:
20a - 3 = 11a + 6
Method 1: - deal with the unknowns first
9a - 3 = 6 (subtract 11a from both sides of the equation)
9a = 9 (add 3 to both sides of the equation) a = 1 (divide both sides by 9)
Method 2: - deal with the numbers first
20a = 11a + 9 (add 3 to both sides of the equation)
9a = 9 (subtract 11a from both sides of the equation) a = 1 (divide both sides by 9)
Again you get the same soution - try both methods and see which you prefer!
I think of a number (lets call it x) and add 6 to it.
The result is equal to twice the number I started with.
What number did I start with?
Write the question into an equation to solve:
x + 6 = 2x (Now this is easy to solve) 6 = x (subtracting x from both sides)
Factorising Expressions
We can rewrite 24 in many ways
24 = 2 x 12
24 = 4 x 6
24 = 12 + 12
24 = 20 + 4
Lets look further into 20 + 4
20 has factors 1,2,4,5,10 and 20
4 has factors 1,2,4
So the HCF of 20 and 4 is 4.
We can no rewrite 24 as:
24 = 20 + 4
24 = 4(5 + 1)
When we have unknown terms in an expression, we can find the highest common factor and take it outside of the bracket, this process is called factorising.
Example:
7x + 14
7 has factors 1,7
14 has factors 1,2,7,14
So the HCF of 7 and 14 is 7.
So we can take 7 outside the bracket as a common factor and rewrite the expression as:
7x + 14 = 7(x + 2)
We can check that we have factorised correctly by expanding out the brackets!
7(x + 2) = 7x + 7 x 2 = 7x + 14
Hurray - we have the correct answer! (Don't forget this as it is a great way to check your answers, especially in an exam)
When we factorise quadratic expressions such as x2 + 3x + 2 we are looking for 2 factors to put into double brackets (x )(x ).
To do this we have to find 2 numbers (our factors) that will multiply together to give us 2 and add toether to give us 3 (relating to the coefficient of x)
2 has factors 1 and 2, so when we multiply these we get 2 and when we add them together we get 3.
We can now say that x2 + 3x + 2 factorises to (x + 1)(x + 2).
Example 2:
x2 + 7x + 12
12 has factors 1 and 12, 2 and 6, 3 and 4.
1 + 12 doesn't equal 7 so these aren't our factors.
2 + 6 doesn't equal 7 so these aren't our factors.
3 + 4 = 7 so we can say that 3 and 4 are our factors and...
x2 + 7x + 12 factorises to (x + 3)(x + 4)
We can easily check this by expanding out the brackets by using either the FOIL or Smiley Face methods from earlier this term. See above to refresh.
Try out some questions on factorising.
Remember your rules for negative numbers as these can get tricky.
-2 x -3 = 6
-2 x 3 = -6
2 x -3 = -6
2 + 3 = 5
2 - 3 = -3 + 2 = -1
-2 + 3 = 3 - 2 = 1
Simplifying Expressions
When simplifying expression we collect like terms so that the expression is reduced to it's simplest form.
Example:
x + x + y = 2x + y
x2 + 2x + 3x + 12 = x2 + 5x + 12
We can't collect terms x2 and x together in the same way that we can't add together an area and a length.
Expanding Brackets
When expanding out brackets we need to make sure that the term outside of the bracket is multiplied by everything inside.a(b + c) = ab + ac This is called the distributive law.
When expanding out double brackets we need to apply the same principle.
(a + b)(c + d) = a(c + d) + b(c + d)
= ac + ad + bc + bd
If the values of a,b,c and d allow then terms can be collected togther to simplify the expression.
There are 2 methods here than I like to use:
FOIL Method - remembering to multiply the First terms, Outside terms, Inside terms and Last terms in the double brackets.
E.g. (a + b)(c + d)
F = ac
O = ad
I = bc
L = bd
Smiley Face Method - create 2 eyebrows, and nose and a mouth to join all the terms, then Multiply along the lines to expand out the brackets.
E.g. (a + b)(c + d)
left eyebrow = ac
right eyebrow = bd
nose = bc
mouth = ad
Solving Linear Equations
An equation is an expression which is equal to something.Example 1
x + 6 = 9
Think of an equation like balanced scales. When we start the equation the scales are balanced. If we are to change anything on one side of the equals sign, we must always do the same thing to the other side for the scales to remain balanced.
x + 6 = 9 (to get x on it's own we want to take 6 away on the LHS, but we must remember to do the same to the RHS)
x + 6 - 6 = 9 - 6
x = 3
When solving an equation, we always line up the equal signs. The practise of Maths is an art and we should respect that in our working out. I just can't get it to line up right on this wiki!
Example 2
4x = 20 (we divide both sides of the equation by 4)
4x = 20
4 4
x = 5
If I went into a shop and bought 6 twix's and a 10p mix for £1.90, how much did each twix cost?
Re write this as an equation to solve it. [twix = t]
6t + 0.10 = 1.90 (firstly we subtract 10p from both sides)
6t + 0.10 - 0.10 = 1.90 - 0.10
6t = 1.80 (then we divide both side by 6)
6t = 1.80
6 = 6
t = 0.30
So one twix equals 30p.
Solving Linear Equations with brackets
We started with the month you were born, lets call this m.
Then we multiplied by 5
Then we added 7 to our answer
Then we multiplied our answer by 4
Then we added 13 to our answer
Then we multiplied by 5
Then we added the day number of our birthday
5m
5m + 7
4(5m + 7) = 20m + 28
20m + 28 + 13 = 20m + 41
5(20m + 41) = 100m + 205
100m + 205 + d = 100m + d + 205
Giving the formula 100m + d + 205
Say we are looking at the 4th of July:
100(July) + 4 + 205 = 100(7) + 4 + 205
704 + 205
Can you see how the trick will always work?
If you give me the number 909
I subtract 205 to give me 704 where 04 is the day and 7 is the month!
Try is out on your friends – can you amaze them with you mathmagic abilities?!
Try your equation solving skills at - http://mathsnet.net/algebra/l1_equation.html
Set 1 you want to be working at level 3/4 but try them and see how well you do!
Solving Equations ...(can get a bit tricky)
When we have tricky equations that look like this:
x + 1 = 7
2
It's not as tricky as you first might think... but make sure you follow the steps and what you do to one side you must do to the other to make sure that the balance stays equal!
This equation is the same as 1/2(x + 1) = 7
We can either expand the brackets out - or to make it easier, as (x+1) is all multiplied by 1/2 we can multiply by 2 to get x + 1 on it's own on the LHS, like this.
1/2(x + 1) = 7
x + 1 = 2 x 7
x + 1 = 14 (now to get x on it's own to solve it we subtract 1 from both sides)
x = 14 - 1
x = 13
We've cracked it! - It's not as scary as it first looks
Give it a try with some of the questions from the lesson and see how well you can solve equations.
Solving Equations often relies on methods you have already learned about simplifying expressions, expanding brackets and collecting like terms.
Example:
2(8x - 7) = 34
Method 1
16x - 14 = 34 (expand out the brackets first)
16x = 48 (add 14 to both sides)
x = 3 (divide both sides by 16)
Method 2
8x - 7 = 17 (divide both sides by 2- we can do this as 2 is multiplied by everything inside the bracket on the left hand side)
8x = 24 (add 7 to both sides)
x = 3 (divide both sides by 8)
Try both method to see that they both work!
What about when we have an unknown on both sides?
Example:
20a - 3 = 11a + 6
Method 1: - deal with the unknowns first
9a - 3 = 6 (subtract 11a from both sides of the equation)
9a = 9 (add 3 to both sides of the equation)
a = 1 (divide both sides by 9)
Method 2: - deal with the numbers first
20a = 11a + 9 (add 3 to both sides of the equation)
9a = 9 (subtract 11a from both sides of the equation)
a = 1 (divide both sides by 9)
Again you get the same soution - try both methods and see which you prefer!
I think of a number (lets call it x) and add 6 to it.
The result is equal to twice the number I started with.
What number did I start with?
Write the question into an equation to solve:
x + 6 = 2x (Now this is easy to solve)
6 = x (subtracting x from both sides)
Factorising Expressions
We can rewrite 24 in many ways
24 = 2 x 12
24 = 4 x 6
24 = 12 + 12
24 = 20 + 4
Lets look further into 20 + 4
20 has factors 1,2,4,5,10 and 20
4 has factors 1,2,4
So the HCF of 20 and 4 is 4.
We can no rewrite 24 as:
24 = 20 + 4
24 = 4(5 + 1)
When we have unknown terms in an expression, we can find the highest common factor and take it outside of the bracket, this process is called factorising.
Example:
7x + 14
7 has factors 1,7
14 has factors 1,2,7,14
So the HCF of 7 and 14 is 7.
So we can take 7 outside the bracket as a common factor and rewrite the expression as:
7x + 14 = 7(x + 2)
We can check that we have factorised correctly by expanding out the brackets!
7(x + 2) = 7x + 7 x 2 = 7x + 14
Hurray - we have the correct answer! (Don't forget this as it is a great way to check your answers, especially in an exam)
Try out your factorising skills here:
Factorising Quadratic Expressions
When we factorise quadratic expressions such as x2 + 3x + 2 we are looking for 2 factors to put into double brackets (x )(x ).
To do this we have to find 2 numbers (our factors) that will multiply together to give us 2 and add toether to give us 3 (relating to the coefficient of x)
2 has factors 1 and 2, so when we multiply these we get 2 and when we add them together we get 3.
We can now say that x2 + 3x + 2 factorises to (x + 1)(x + 2).
Example 2:
x2 + 7x + 12
12 has factors 1 and 12, 2 and 6, 3 and 4.
1 + 12 doesn't equal 7 so these aren't our factors.
2 + 6 doesn't equal 7 so these aren't our factors.
3 + 4 = 7 so we can say that 3 and 4 are our factors and...
x2 + 7x + 12 factorises to (x + 3)(x + 4)
We can easily check this by expanding out the brackets by using either the FOIL or Smiley Face methods from earlier this term. See above to refresh.
Try out some questions on factorising.
Remember your rules for negative numbers as these can get tricky.
-2 x -3 = 6
-2 x 3 = -6
2 x -3 = -6
2 + 3 = 5
2 - 3 = -3 + 2 = -1
-2 + 3 = 3 - 2 = 1
Extra notes...
Factorising Quadratics for 2x2 + ax + b
When we factorise quadratics such as 2x2 + 5x + 2 we have to be more careful as to the factors we choose.
We can see that our brackets will look like (2x )(x )
2 has the factors 1 and 2, but we then have to check that these work by putting them into the formula (F1 + (2 x F2))
Again we can check this by expanding out the brackets!
Test your self on these using the tables to help you calculate: